To determine the domain and range of the function [tex]\( C(m) \)[/tex], which describes the total cost of Haley's monthly cell phone bill based on her minutes of usage, we start with the given function:
[tex]\[ C(m) = 0.05m + 20 \][/tex]
### Domain
The domain of a function refers to the set of all possible input values (in this case, the values of [tex]\( m \)[/tex] which represent the number of minutes used).
1. Since [tex]\( m \)[/tex] denotes the number of minutes, it cannot be negative. Haley cannot use a negative number of minutes.
2. Therefore, the smallest value that [tex]\( m \)[/tex] can take is 0 (indicating zero minutes used).
Thus, the domain of the function is:
[tex]\[ m \geq 0 \][/tex]
This means that the domain includes all non-negative real numbers.
### Range
The range of a function refers to the set of all possible output values (in this case, the values of [tex]\( C \)[/tex] which represent the total cost).
1. To find the range, consider the function [tex]\( C(m) = 0.05m + 20 \)[/tex] again.
2. Notice that the smallest value [tex]\( m \)[/tex] can take is 0 (from the domain). Plugging [tex]\( m = 0 \)[/tex] into the function:
[tex]\[ C(0) = 0.05 \cdot 0 + 20 = 20 \][/tex]
This gives us the lowest possible value of [tex]\( C \)[/tex], which is [tex]$20.
3. As \( m \) increases, \( C(m) \) increases linearly because \( C(m) \) is a linear function with a positive slope (0.05). There is no upper limit to the number of minutes \( m \) that can be used, and thus no upper limit to the cost \( C(m) \).
Thus, the range of the function is:
\[ C \geq 20 \]
This means the range includes all real numbers greater than or equal to $[/tex]20.
### Summary
- Domain: [tex]\( m \geq 0 \)[/tex] (non-negative minutes)
- Range: [tex]\( C \geq 20 \)[/tex] (the cost will always be at least $20)
So, the domain [tex]\( m \geq 0 \)[/tex] and the range [tex]\( C \geq 20 \)[/tex] correctly describe the function [tex]\( C(m) = 0.05m + 20 \)[/tex].
